What is the Hammerstad-Jensen model and when is it accurate for microstrip impedance calculation?
Hammerstad-Jensen Microstrip Model
The Hammerstad-Jensen equations were published in 1980 and remain the standard analytical model for microstrip impedance. They solve the electrostatic problem of the microstrip cross-section using conformal mapping techniques with empirical corrections for conductor thickness. The resulting formulas are computationally fast and surprisingly accurate across a wide range of geometries.
| Parameter | Semi-Rigid | Conformable | Flexible |
|---|---|---|---|
| Loss (dB/m at 10 GHz) | 0.8-2.5 | 1.0-3.0 | 1.5-5.0 |
| Phase Stability | Excellent | Good | Fair |
| Bend Radius | Fixed after forming | Hand-formable | Continuous flex OK |
| Shielding (dB) | >120 | >90 | >60-90 |
| Cost (relative) | 2-5x | 1.5-3x | 1x |
Cable Selection Criteria
The model calculates two key parameters: the characteristic impedance (Z0) and the effective dielectric constant (εeff). From these, all other transmission line parameters can be derived: propagation velocity = c/√εeff, wavelength = λ0/√εeff, and the synthesis problem (finding W/h for a desired Z0) can be solved by inverting the equations.
- Performance verification: confirm specifications against the application requirements before finalizing the design
- Environmental factors: temperature range, humidity, and vibration affect long-term reliability and parameter drift
- Cost vs. performance: evaluate whether the application demands premium components or standard commercial grades
- Interface compatibility: verify impedance, connector type, and mechanical form factor match the system architecture
- Margin allocation: include sufficient design margin to account for manufacturing tolerances and aging effects
Loss and Phase Stability
Limitations of the model include the assumption of zero-thickness ground plane, infinite substrate extent, no enclosure effects, and static (frequency-independent) behavior. At frequencies where the substrate thickness approaches λ/10, the quasi-static assumption breaks down, and full-wave effects (dispersion, higher-order modes, radiation) must be considered separately using frequency-dependent correction models.
Frequently Asked Questions
Is it accurate enough for production design?
For initial design and hand calculations: yes, below 10 GHz. For production designs, always verify with electromagnetic simulation (Momentum, Sonnet, HFSS) because the model does not account for manufacturing-specific effects like etch taper, copper roughness, solder mask, and adjacent via effects.
What about synthesis - finding W for a given Z0?
The inverse problem (finding W/h for a desired Z0 and εr) can be solved using Wheeler's synthesis formulas or by numerically inverting the Hammerstad-Jensen analysis equations. Most RF design tools include built-in microstrip synthesis calculators based on these models.
How does it compare to EM simulation?
For a straight, uniform microstrip line, the Hammerstad-Jensen model agrees with EM simulation to within 1-2% up to about 10 GHz. The discrepancy increases at higher frequencies due to dispersion and at very narrow or very wide traces due to higher-order mode effects. EM simulation is always more accurate but takes seconds to minutes versus microseconds for closed-form equations.